How can I get my friend to overcome the fear of other people knowing she has a problem and going to aa?
You can't. You can help her by being supportive and non-judgmental, by being a friend, but you cannot "get her" to do anything really. There are "open" aa meetings where people with and without alcoholism, and people who are just curious can go. Offer to go to an open meeting with her. You could ask one of the older members to talk to her and even be a temporary sponsor. You can go to alanon yourself.
Tell her: "everybody knows you have a problem now. When you go to aa, they will know you have a solution.".
Fear of the unknown. Is common. The choice is hers as to whether she desires to go to a 12 step meeting. However suspect that her fear would begin diminishing once she attended a meeting.